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Fearless Symmetry by Robert Gross, Avner Ash

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RETROSPECT 263
Peering Past the Frontier
As we move on to other systems of Z-equations, we find that
particular examples tend to be less and less of interest. Do you
really care exactly how many solutions to the system
33x
2
42xy + z
72
= 445
x
2
34xyz + z
333
= 12
x + y
2
+ 2z
3
= 11
there are with x, y, and z all integers? Unless this problem has
some burning practical significance, it is not very interesting
merely on its own. We just made it up at random. On the other
hand, it does represent the kind of problem that interests us in
general, namely, finding the solutions to a system of Z-equations.
Any adequate theory we develop should eventually be able to tell
us about this random system and all other systems too.
Thus, our focus tends to shift to whole classes of problems, and
therefore to the structural or qualitative statements we can make
or prove about them. For example, what computable properties of
a system of Z-equations can guarantee that there are only finitely
many solutions with the variables taking on only rational values?
Such questions, which are at the frontier of research, lead back to
more structural questions about varieties and the absolute Galois
group of Q.
For another example, consider the statement: The absolute
Galois group of Q is big. Now, make it into a precise assertion and
prove it. That might include proving that for any positive integer n
and any prime p, there is a Galois representation whose image
2
is
all of GL(n, F
p
). (It is not known if this is true.) But there will be
many other ways of envisioning the “bigness” of the absolute Galois
group of Q, according to various other aspects of its structure.
In this way, after we leave behind particular Z-equations or sys-
tems that have provoked our study in the first place, the structures
2
The image of a group representation φ is the set of all elements in the target that are
actually values of φ.
264 CHAPTER 23
themselves—the absolute Galois group of Q, for example—tend to
take center stage. As the structures become more understood, par-
ticular problems—the congruent number problem, for example—
will achieve their solution. This dialectic between the general and
the specific is very common in the history of mathematics.
Thus, the focus of theoretical interest in the study of systems
of Z-equations tends to shift, as the theory is developed, to Galois
groups and their representations. We begin to ask questions: How
big can representations of the Absolute Galois group of Q be? How
can certain families of such representations be parametrized? But if
for any reason we need to say something about a particular system
of Z-equations, the theory should stay closely enough connected
to the motivating Diophantine problems to be a powerful tool for
attacking these problems.

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